Mathematical Sciences: Analysis of the Quasi-Regular Representation of Lie Groups and Index Theory on Non-CompactManifolds

数学科学：李群拟正则表示分析和非紧流形指标论

• 批准号: 8703572
• 负责人: Jeffrey Fox
• 金额: $ 2.78万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-01 至 1989-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8703572&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Analysis; Quasi; Regular

One of the most fundamental developments in modern science has been the theory of quantum mechanics. The attempt to understand both the physics and the underlying mathematics has led directly to current frontiers in two important areas of Modern Analysis, representation theory and operator algebras. Representation theory is a means of exploiting the inherent physical symmetries, while the concept of operator algebras was invented to provide the correct framework for quantization. Recently, there has been a great expansion in our understanding of the mathematics underlying quantization. This has come from deep interactions between physical ideas and mathematical constructs. In particular, representation theory and operator algebras have been brought together recently in a powerful theory that centers around the celebrated Atiyah-Singer index theorem. Professor Fox is a young researcher who has mastered the broad range of mathematics necessary to contribute to this field. He is an expert in representation theory, operator algebras, and index theory. He proposes to develop an index theorem for noncompact manifolds that generalizes Connes' extension of the Atiyah-Singer index theorem for compact manifolds. This index theorem would be used to study the discrete spectrum of the quasi-regular representation of a semi-simple Lie group acting on a locally symmetric space of finite volume. In addition to its intrinsic interest in representation theory and operator algebras, a solution to this problem would lead to a better understanding of the various facets of quantization. The investigator's prior work, in part collaborative, has laid a solid foundation for undertaking this study.

现代科学最基本的发展之一是量子力学理论。对物理学和基本数学的理解的尝试直接导致了现代分析的两个重要领域--表示理论和算子代数--的当前前沿。表示理论是一种利用内在物理对称性的方法，而算子代数的概念是为了提供量子化的正确框架而发明的。最近，我们对量子化背后的数学原理的理解有了很大的扩展。这来自于物理思想和数学结构之间的深度互动。特别是，表示理论和算子代数最近在一个强大的理论中被结合在一起，该理论以著名的Atiyah-Singer指数定理为中心。福克斯教授是一名年轻的研究人员，他掌握了为这一领域做出贡献所需的广泛数学知识。他是表示论、算子代数和指数理论方面的专家。他提出了一个非紧流形的指标定理，推广了Connes关于紧流形的Atiyah-Singer指标定理。该指标定理可用于研究有限体积局部对称空间上的半单李群的拟正则表示的离散谱。除了它对表示理论和算子代数的内在兴趣之外，解决这个问题将导致对量子化的各个方面的更好的理解。研究人员先前的工作，部分是合作的，为开展这项研究奠定了坚实的基础。